Shared randomness and device-independent dimension witnessing

TL;DR

This paper investigates how shared randomness influences the ability to witness the dimension of uncharacterized systems in device-independent scenarios, revealing fundamental differences in classical and quantum resources and the structure of probability sets.

Contribution

It provides a detailed analysis of the role of shared randomness in dimension witnessing, highlighting its impact on the structure and robustness of these witnesses in classical and quantum contexts.

Findings

Shared randomness can make classical systems mimic quantum behaviors of higher dimension.

Sets of low-dimensional behaviors are negligible without shared randomness, unlike with it.

Presence of shared randomness significantly alters the structure and properties of the sets of probability distributions.

Abstract

It has been shown that the conditional probability distributions obtained by performing measurements on an uncharacterized physical system can be used to infer its underlying dimension in a device-independent way both in the classical and quantum setting. We analyze several aspects of the structure of the sets of probability distributions corresponding to a certain dimension taking into account whether shared randomness is available as a resource or not. We first consider the so-called prepare-and-measure scenario. We show that quantumness and shared randomness are not comparable resources. That is, on the one hand, there exist behaviours that require a quantum system of arbitrarily large dimension in order to be observed while they can be reproduced with a classical physical system of minimal dimension together with shared randomness. On the other hand, there exist behaviours which require exponentially larger dimensions classically than quantumly even if the former is supplemented with shared randomness. We also show that in the absence of shared randomness, the sets corresponding to a sufficiently small dimension are negligible (zero-measure and nowhere dense) both classically and quantumly. This is in sharp contrast to the situation in which this resource is available and explains the exceptional robustness of dimension witnesses in the setting in which devices can be taken to be uncorrelated. We finally consider the Bell scenario in the absence of shared randomness and prove some non-convexity and negligibility properties of these sets for sufficiently small dimensions. This shows again the enormous difference induced by the availability or not of this resource.